Valentino Tosatti

Columbia University
Department of Mathematics, Room 600
2990 Broadway, New York, NY 10027
Email: tosatti@math.columbia.edu
Phone: 212-854-1592
Fax: 212-854-8962

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I am a Ritt Assistant Professor at Columbia University. Before coming to Columbia I was a graduate student at the Department of Mathematics, Harvard University, where I got my Ph.D. on June 2009. My advisor was Shing-Tung Yau. Before that, I got an Italian Laurea degree at the University of Pisa and at the Scuola Normale Superiore, under the supervision of Marco Abate.

I do research in Differential Geometry, Geometric Analysis, Complex Geometry and Partial Differential Equations. Some specific topics are: Kähler Geometry, Calabi-Yau manifolds, almost-complex geometry, geometric flows, complex Monge-Ampère equations.

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Teaching

Calculus IV - Spring 2010

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Publications and Preprints

• (with G. Székelyhidi) Regularity of weak solutions of a complex Monge-Ampère equation
  [arXiv] preprint 2009
  
  
• (with B. Weinkove) The complex Monge-Ampère equation on compact Hermitian manifolds
  [arXiv] preprint 2009
  
  
• (with B. Weinkove) Estimates for the complex Monge-Ampère equation on Hermitian and balanced manifolds
  [arXiv] to appear in Asian J. Math. 2010
  
  
• (with B. Weinkove) The Calabi-Yau equation on the Kodaira-Thurston manifold
  [arXiv] preprint 2009
  
  
• Adiabatic limits of Ricci-flat Kähler metrics
  [arXiv] to appear in J. Differential Geom. 2010
  
  
• (with B. Weinkove)The Calabi-Yau equation, symplectic forms and almost complex structures
  [arXiv] to appear in Geometric Analysis: Present and Future, International Press, 2010.
  
  
• Kähler-Ricci flow on stable Fano manifolds
  [arXiv] [journal] J. Reine Angew. Math. 2010.
  
  
• Limits of Calabi-Yau metrics when the Kähler class degenerates
  [arXiv] [journal] J. Eur. Math. Soc. (JEMS) 11 (2009), no.4, 755-776.
  
  
• (with B. Weinkove and S.-T. Yau) Taming symplectic forms and the Calabi-Yau equation
  [arXiv] [journal] Proc. London Math. Soc. 97 (2008), no.2, 401-424.
  
  
• A general Schwarz Lemma for almost-Hermitian manifolds
  [arXiv] [journal] Comm. Anal. Geom. 15 (2007), no.5, 1063-1086.
  
  
• (with B. Weinkove) The Calabi Flow with small initial energy
  [arXiv] [journal] Math. Res. Lett. 14 (2007), no.6, 1033-1039.
  
  
• On the Critical Points of the E_k Functionals in Kähler Geometry
  [arXiv] [journal] Proc. Amer. Math. Soc. 135 (2007), no.12, 3985-3988.
  
  
• (with L. Pietronero, E. Tosatti and A. Vespignani) Explaining the uneven distribution of numbers in nature
  [arXiv] [journal] Phys. A, 293 (2001), 297-304.

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Here's my CV, updated in December 2009.

Finally you might be interested in some other mathematical stuff.

• Geometry of complex Monge-Ampère equations
  [PDF] 4 June 2009, PhD Thesis, Harvard University, Advisor: Prof. Shing-Tung Yau.

• Extremal Sobolev inequalities and applications
  [PDF] 12 March 2005, Minor Thesis, Harvard University, Advisor: Prof. Ben Weinkove.

• Index theorems for morphisms of vector bundles and foliations
  [PDF] (in Italian) 10 June 2004, Laurea Thesis, University of Pisa, Advisor: Prof. Marco Abate.

• Uniqueness of CPⁿ
  [PDF] 30 October 2007, an exposition of a theorem of Hirzebruch, Kodaira, Yau.

• Lower bound of the K-energy on small deformations of cscK manifolds
  [PDF] 19 November 2009, an exposition of a theorem of Chen.

Schedule for Columbia's Informal Complex Geometry and PDE Seminar.
Schedule for Harvard's Yau's students seminar.

My Erdős number is 3.